Structure and Reactivity in Aqueous Solution. Characterization of Chemical and Biological Systems 9780841229808, 9780841214682, 0-8412-2980-5

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ACS SYMPOSIUM SERIES 568

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Structure and Reactivity in Aqueous Solution Characterization of Chemical and Biological Systems

Christopher J. Cramer, EDITOR University of Minnesota

Donald G. Truhlar, EDITOR University of Minnesota

Developed from a symposium sponsored by the Division of Physical Chemistry at the 207th National Meeting of the American Chemical Society, San Diego, California, March 13-18, 1994

American Chemical Society, Washington, DC 1994

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

QD 540 .S77 1994 Copy 1

Structure and r e a c t i v i t y i n aqueous solution

Library of Congress Cataloging-in-Publication Data Structure and reactivity in aqueous solution: characterization of chemical and biological systems / Christopher J. Cramer, editor; Donald G. Truhlar, editor.

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p.

cm.—(ACS symposium series, ISSN 0097-6156; 568)

"Developed from a symposium sponsored by the Division of Physical Chemistry at the 207th National Meeting of the American Chemical Society, San Diego, California, March 13-18, 1994." Includes bibliographical references and indexes. ISBN 0-8412-2980-5 1. Solution (Chemistry)—Congresses. I. Cramer, Christopher J., 1961- . II. Truhlar, Donald G., 1944III. American Chemical Society. Division of Physical Chemistry. IV. Series. QD540.S77 1994 541.3'422-dc20

94-29430 CIP

The paper used in this publication meets the minimum requirements of American National Standard for Information Sciences—Permanence of Paper for Printed Library Materials, ANSI Z39.48-1984. Copyright © 1994 American Chemical Society All Rights Reserved. The appearance of the code at the bottom of the first page of each chapter in this volume indicates the copyright owner's consent that reprographic copies of the chapter may be made for personal or internal use or for the personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per-copy fee through the Copyright Clearance Center, Inc., 27 Congress Street, Salem, MA 01970, for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to copying or transmission by any means—graphic or electronic—for any other purpose, such as for general distribution, for advertising or promotional purposes, for creating a new collective work, for resale, or for information storage and retrieval systems. The copying fee for each chapter is indicated in the code at the bottom of thefirstpage of the chapter. The citation of trade names and/or names of manufacturers in this publication is not to be construed as an endorsement or as approval by ACS of the commercial products or services referenced herein; nor should the mere reference herein to any drawing, specification, chemical process, or other data be regarded as a license or as a conveyance of anyrightor permission to the holder, reader, or any other person or corporation, to manufacture, reproduce, use, or sell any patented invention or copyrighted work that may in any way be related thereto. Registered names, trademarks, etc., used in this publication, even without specific indication thereof, are not to be considered unprotected by law. PRINTED IN THE UNITED STATES OF AMERICA

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

1994 Advisory Board ACS Symposium Series

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M . Joan Comstock, Series Editor Robert J. Alaimo Procter & Gamble Pharmaceuticals

Douglas R. Lloyd The University of Texas at Austin

Mark Arnold University of Iowa

Cynthia A. Maryanoff R. W. Johnson Pharmaceutical Research Institute

David Baker University of Tennessee Arindam Bose Pfizer Central Research

Julius J. Menn Western Cotton Research Laboratory, U.S. Department of Agriculture

Robert F . Brady, Jr. Naval Research Laboratory

Roger A. Minear University of Illinois at Urbana-Champaign

Margaret A. Cavanaugh National Science Foundation

Vincent Pecoraro University of Michigan

Arthur B. Ellis University of Wisconsin at Madison

Marshall Phillips Delmont Laboratories

Dennis W. Hess Lehigh University

George W. Roberts North Carolina State University

Hiroshi Ito IBM Almaden Research Center

A. Truman Schwartz Macalaster College

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Lawrence P. Klemann Nabisco Foods Group Gretchen S. Kohl Dow-Corning Corporation Bonnie Lawlor Institute for Scientific Information

L. Somasundaram DuPont Michael D. Taylor Parke-Davis Pharmaceutical Research Peter Willett University of Sheffield (England)

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

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Foreword THE ACS

S Y M P O S I U M SERIES was first published in 1974

to

provide a mechanism for publishing symposia quickly in book form. The purpose of this series is to publish comprehensive books developed from symposia, which are usually "snapshots in time" of the current research being done on a topic, plus some review material on the topic. For this reason, it is necessary that the papers be published as quickly as possible. Before a symposium-based book is put under contract, the proposed table of contents is reviewed for appropriateness to the topic and for comprehensiveness of the collection. Some papers are excluded at this point, and others are added to round out the scope of the volume. In addition, a draft of each paper is peer-reviewed prior to final acceptance or rejection. This anonymous review process is supervised by the organizers) of the symposium, who become the editor(s) of the book. The authors then revise their papers according to the recommendations of both the reviewers and the editors, prepare camera-ready copy, and submit the final papers to the editors, who check that all necessary revisions have been made. As a rule, only original research papers and original review papers are included in the volumes. Verbatim reproductions of previously published papers are not accepted. M. Joan Comstock Series Editor

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

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Preface W A T E R IS A CRITICAL C O M P O N E N T of the earth's surface and of the biological organisms upon it. A l l biochemical reactions take place in water, and water's enthalpic and entropie characteristics are unique. Consequently, aqueous solutions are a focal point for studies of structure and reactivity. Participants in the symposium from which this book was developed came from around the globe to engage in stimulating exchanges highlighting recent research in this area. Speakers hailed from Australia, Belgium, Canada, France, Italy, the Netherlands, Spain, and the United States. The breadth of the experimental and theoretical chemistry discussed was reflected in the diverse affiliations of the speakers, with representatives from academia, industry, and government laboratories all presenting their most exciting results. Several sessions attracted audiences that overflowed the 200-seat capacity meeting room, a testimony to the high caliber of the science discussed This book represents a remarkable recording of those four days. Of the 28 invited and five contributed lectures, book chapters were developed from 24 of the former and two of the latter. In addition, one chapter derives from the work of a researcher in the field who was unable to travel to the symposium. In an effort to keep this material fresh and exciting, we imposed draconian deadlines on all of our authors and peer reviewers and in general harassed them to the limits of human endurance. Thanks to their good humor and universal accommodation, this volume was completed a mere seven weeks after the end of the meeting! We take this opportunity to thank all of the participants in our symposium and we look forward to future gatherings devoted to this subject area. CHRISTOPHER J. CRAMER AND DONALD G . TRUHLAR Department of Chemistry and Supercomputer Institute 207 Pleasant Street, S.E. University of Minnesota Minneapolis, M N 55455-0431 May 12, 1994

ix

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

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Sara and Stephanie and William and Matthew

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

Chapter 1

Structure and Reactivity in Aqueous Solution An Overview Christopher J. Cramer and Donald G. Truhlar

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Department of Chemistry and Supercomputer Institute, University of Minnesota, 207 Pleasant Street, S.E., Minneapolis, MN 55455-0431

This introductory chapter provides a brief overview of the current state of the art in understanding and modeling structure and reactivity in aqueous solution. The contents of the chapters found in this book are discussed and areas where theory and experiment are working in tandem are highlighted. Water is a remarkable substance. It covers two thirds of the Earth's surface, it makes up a large fraction of the total mass of biological organisms, it serves as the solvent in which essentially all biochemical reactions take place, and it's enthalpic and entropie characteristics repeatedly set it apart from other liquids (1). It is thus hardly surprising that chemists with an interest in the role that solvent plays in determining structure and reactivity devote themselves to aqueous solutions more than any other kind. For theorists in particular this has been the case. Relative to modeling in the gas phase, it is only recently that theoretical techniques capable of simulating condensed phase chemistry have been developed. There are two complementary approaches which may be taken to modeling a solute embedded in a solvent. One is the quantum mechanical continuum approach (2-6) where the solvent is replaced by a continuous medium having the same dielectric constant as bulk liquid. This is a mean-field approach in which the solute polarizes the bulk medium, which backpolarizes the solute, etc.—a physical picture dating back to Onsager (7-9). In Chapter 2, the historical development of one such model is detailed by Tomasi, with special attention paid to algorithmic details. In addition, the present directions in which the model is expanding are detailed, including techniques to separate the continuum into fast and slow components for dynamical studies and methods for incorporating local anisotropy into the continuum model. One significant drawback of any pure continuum approach is that the solvent bulk dielectric constant does not accurately describe the electric polarization field right up to the molecular "surface". Instead, the interaction of a solute with at least the first solvation shell is typically characterized by specific local energy components. One method which has been explored to address this is the assignment

0097-6156/94/0568-0001$08.00/0 © 1994 American Chemical Society

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of local surface tensions to specific regions of the molecular surface (10-13), Chapter 3 by Storer et al. describes continuum solvation models incorporating such local effects. In addition, it provides specific details of the algorithms required to accurately account for dielectric screening, an important effect whereby the interaction of one portion of the solute with the surrounding continuum is mediated by the intervention of remaining portions of the solute. Finally, this chapter presents an example of using Specific Range Parameters for the modeling of particular chemical reactions, in particular for the study of the effect of solvation on the Claisen rearrangement, a reaction that several other chapters in this symposium also discuss (vide infra). The interaction energy of a set of charges with a surrounding dielectric medium may be found by solution of the Poisson equation (14-17). When the continuous charge density of a molecular solute is replaced by a collection of point charges, the solution of the Poisson equation becomes more facile; however, there are ambiguities involved in modeling a continuous charge density with point charges. There are also inherent ambiguities in defining the dielectric boundary between the solute and the solvent. In Chapter 4 Lim et al. discuss the use of calculated quantum mechanical densities for the evaluation of these quantities, and they present results for the solvation free energies of a number of ions and organic molecules. Tawa and Pratt present a similar Poisson equation derived formalism in Chapter 5, where they calculate the aqueous solvent-induced potential of mean force for the dissociation of sodium chloride and for nucleophilic substitution and addition reactions. Tawa and Pratt compare their calculations to other models where solvent has not been replaced by the mean-field continuum. This alternative approach, i.e., representing the solvent in its discrete, molecular form, allows investigation of the microscopic details of solvation. It has been the case so far, though, that the number of solvent molecules that must be included to adequately model bulk solvation for a given solute is so large that a quantum mechanical treatment of the system has been prohibitively difficult. Instead, the solute-solvent and solvent-solvent interactions have been modeled classically using empirical force fields (18-24) or—very recently—a combination of such a force field with terms representing polarization of the water molecules. By using such a force field and following the trajectory of the system over time (2528), it is possible to calculate dynamical properties or time-averaged equilibrium properties, or Monte Carlo methods (22,29) may be used to obtain equilibrium properties. Clementi and Corongiu discuss in Chapter 7 the step-by-step development of a force field that has been designed to reproduce numerous static and dynamical properties of flexible water over a sizable temperature range. Nyberg and Haymet analyze in Chapter 8 a more complete model for liquid water in which the force field permits dissociation (accompanied by ionization) of individual water molecules. They compare their results to other models with respect to predicting the pH of liquid water. Finally, in Chapter 22 Van Belle et al. present calculations with polarizable water molecules (vide infra). Nonequilibrium solvation can be critical for many processes, e.g., electron transfer (30) and spectroscopy (31). If the effects of solvation can be modeled as an extra, general coordinate on a solute potential energy surface, it becomes

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

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straightforward to extend dynamical studies from the gas-phase into solution (3235). Using an approach which permits the separation of equilibrium and nonequilibrium components of aqueous solvation, Garrett and Schenter examine in Chapter 9 the radical addition of hydrogen atoms to benzene. In particular, they use variational transition state theory (36) to calculate kinetic isotope effects (KIEs) and explore solvent-induced changes in the KIEs for deuterium and muonium. Ando and Hynes also adopt this generalized coordinate approach in Chapter 10, where they consider the aqueous ionization of hydrochloric acid. In particular, they assess the importance of nonequilibrium microsolvation in the proton transfer from HC1 to a water molecule and the subsequent separation of the two ions. This same system is also studied by Rivail et al. in Chapter 11 using a quantum mechanical continuum model with either one or two explicit water molecules. In addition to providing this stimulating study which the reader may compare to the work of Ando and Hynes, Rivail et al. also explore the hydrolysis of formamide in aqueous solution, again focusing on the details of those specific water molecules that are not part of the bulk solvent but are instead involved in the reaction itself. A third subject addressed in this chapter is a comparison of the properties of an isolated water molecule, a water molecule in the water dimer, and water in the bulk. These various methods for modeling aqueous solvation provide considerable flexibility to researchers probing specific chemical problems. One area of particular interest is understanding how aqueous solvation affects organic reactions (37,38). In Chapter 12, Bertrdn et al. use various solvation models to study the effect of solvation on several organic reactions, paying particular attention to the location of the transition state along the solvated reaction coordinate and the degree to which nonequilibrium solvation effects must be included in solvation modeling. Ultimately, some of the most interesting organic reactions occurring in aqueous media are those involving biological macromolecules. In this latter area, Warshel has explored many models for aqueous solvation (27,39) as it affects enzyme-mediated reactions. In Chapter 6, Warshel and Chu summarize their experience with numerous macroscopic (e.g., continuum), microscopic (e.g., discrete), and microscopic/macroscopic hybrid water models. In particular, they consider the balance between quantum and classical mechanics strategies and examine some of the algorithmic details of model implementation. One possibility discussed by Warshel involves the replacement of explicit solvent molecules with pseudopotentials. Gordon and co-workers have also been active in this area (40)—their approach divides microsolvated clusters into a "solute" (potentially supermolecular) and a set of water molecules with which it interacts. The solute is treated quantum mechanically; the "spectator" region is modeled using pseudopotentials developed to accurately reproduce electrostatic, polarization, and exchange repulsion interactions in prototypical systems. The marriage of quantum mechanical and classical mechanical treatments, where the former is applied to the solute and the latter to the solvent, is also a subject of much interest (27,41-44). In Chapter 15, Gao applies such a strategy to the Claisen rearrangement, already discussed in Chapter 3, and to the Menshutkin nucleophilic substitution reaction. In addition, Gao extends a classical water model

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into the supercritical regime in order to examine the change in the potential of mean force for ion-pairing in sodium chloride. Tucker and Gibbons also consider supercritical water in Chapter 14, where they consider the hydrolysis of anisole in supercritical solvent They illustrate that clustering of the solvent is an important phenomenon such that local dielectric constants may deviate significantly from the bulk value above the critical point. The selectivity of biologically important reactions, like those mediated by enzymes, depends in part on the ability of macromolecules to recognize the structure of their often complicated substrates (45). As such, it is of considerable interest to examine how aqueous solvation affects conformational equilibria. Venanzi et al. consider this issue for a diuretic acylguanidine, amiloride, in Chapter 18. They additionally extend their studies into the regime of molecular recognition and present results for the hydrolysis of phenyl acetate in aqueous solution both as a free substrate and as a guest in a β-cyclodextrin host The contribution by Wilcox et al, chapter 19, also considers molecular recognition, and it includes a description of their synthetic design of cyclic polyaromatic receptors capable of transporting hydrophobic substrates into aqueous solution. In particular, they provide microscopic and thermodynamic analyses of host-guest interactions and illustrate the synergy between synthesis, spectroscopy, and molecular modeling in their experimental design. Gajewski and Brichford, in Chapter 16, also combine experimental data and statistical modeling in an examination of the effects of solvation on the Claisen rearrangement Their combination of kinetic isotope effect measurements and factor analysis using a variety of solvent descriptors provides a unique perspective which may be compared to the theoretical modeling of Chapters 3 and 15. Severance and Jorgensen provide a fourth perspective on the effect of solvation on the Claisen rearrangement. In Chapter 17, they employ Monte Carlo statistical mechanics simulations (29) with a classical water model. In particular, they consider in detail the effects of multiple conformational minima for the reactant allyl vinyl ether and examine the microscopic solvation of the gas-phase reaction coordinate. Comparison of these four studies of the Claisen rearrangement gives rise to a detailed understanding of the reaction as it occurs in water and moreover serves to illustrate the individual strengths and weaknesses of the various methods employed. This comparison is summarized in Chapter 3. In Chapter 20, Breslow considers the effect of aqueous solvation on a different pericyclic process, the Diels-Alder reaction. This chapter discusses the experimental rates for several Diels-Alder reactions in the presence of various salts designed to tighten or disrupt internal water structure. In addition, other organic reactions and molecular recognition events are examined under the same conditions. Blokzijl and Engberts, in Chapter 21, offer additional experimental insights into aqueous acceleration of the Diels-Alder reaction by comparing interand intramolecular variants of the cycloaddition. They examine in detail the enthalpic and entropie components of the acceleration and identify both the reduction in hydrophobic surface area and the possibility of enhanced solutesolvent hydrogen bonding in the transition state as being critical to the observed rate accelerations.

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

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As part of their explanation for aqueous acceleration of the Diels-Alder reaction, Breslow and Blokzijl and Engberts invoke the tendency of non-polar solutes to associate in order to minimize their exposed surface area in aqueous solution (46-48). In Chapter 22, Van Belle et al. examine this hydrophobic interaction in detail using a simulation model that includes the polarization of discrete water molecules. They examine in particular the potential of mean force for the association of two methane molecules in water and find no evidence for a solvent-separated minimum, in disagreement with several prior studies that did not account for polarizability of the water molecules. Another particularly interesting result of their study is the reduced dipole moment of water molecules within the first two solvation shells of the methane solutes. This result has important implications for the dielectric continuum models discussed elsewhere in this volume. In Chapter 23, Hermann examines the interaction of hydrocarbons in aqueous solution using a decomposition of the free energy of solvation into a cavity-surface-tension contribution and a contribution calculated from configurationally averaged solute-solvent interaction potentials. In addition, comparisons between explicit-solvent molecular dynamics calculations and the results from transferable fragment solvent-distribution functions are offered. The hydrophobic effect manifests itself for situations other than hydrocarbon-water and hydrocarbon-hydrocarbon interactions. Importantly, it appears to play an important role in dictating the folding and higher-order structure of proteins (49,50). Understanding and predicting the structure and dynamics of proteins is of great interest. In Chapter 13, Gai et al. provide experimental details of the aqueous photophysics of 7-azaindole, a potentially useful surrogate chromophore for tryptophan which would permit the observation of short-time scale dynamics when incorporated into enzymes. Scheraga discusses in Chapter 24 theoretical approaches to predicting protein structure. In particular, continuum solvation techniques employing terms depending on either solvent shell volume or solvent-accessible surface area are presented; calculated and measured thermodynamic values are compared for the interaction of individual amino-acid residues and generic organic functionalities. Using molecular simulations, BenNaim also considers the interaction of specific protein residues in Chapter 25. This chapter concludes that solvent-induced attraction between hydrophilic groups can be stronger than that between hydrophobic groups, suggesting that surface tension models which are based on the positive values observed for hydrocarbons may be insufficiently flexible to model the solvation of more complex functional groups. Chapter 26 describes the modeling of a different kind of biopolymer in water, namely D N A ; Beveridge et al. describe a full nanosecond simulation of a dodecamer double helix in aqueous solution and analyze the effect of solvation and of complexation with a repressor-operator protein on the structural dynamics. They find a number of structures which exhibit metastability over periods of hundreds of picoseconds, suggesting that the dangers of short simulation times for biomolecular systems are significant. In Chapter 27 Pohorille and Wilson also explore a large biologically relevant system. They use molecular simulation to study the interaction of a model dipeptide with a glycerol 1-monooleate bilayer. The degree to which the dipeptide penetrates the interfacial region between the solvent and the bilayer is

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explored, as are the differential dynamics of the dipeptide surrounded by water compared to inside the bilayer. Finally, Chapter 28 provides an additional perspective on an interfacial system; Benjamin uses molecular simulation to analyze the dynamics of electron transfer from liquid water to an adjoining 1,2-dichloroethane phase. The overall relaxation dynamics in response to a photochemically induced electron transfer event are found to depend on each of the two solvents and their different respective time scales. In summary, aqueous solvation plays an important role in diverse systems of great chemical importance. This volume attempts to provide as wide a perspective as possible (in a four-day symposium) in this regard. We have reached a stage where theory and experiment may be used in conjunction, complementing each other's strengths and weaknesses, so as to illuminate chemical phenomena at a greater level of detail than would be possible from either one alone. We sincerely hope that some of the examples of this synergy that are provided in this book will stimulate further collaborations in this regard. Literature Cited (1) Reichardt, C. Solvents and Solvent Effects in Organic Chemistry; VCH: New York, 1990. (2) Rivail, J.-L.; Rinaldi, D.Chem.Phys. 1976, 18, 233. (3) Tapia, O. In Quantum Theory of Chemical Reactions; R. Daudel, A. Pullman, L. Salem and A. Viellard, Eds.; Reidel: Dordrecht, 1980; Vol. 2; p. 25. (4) Miertus, S.; Scrocco, E.; Tomasi, J. Chem. Phys. 1981, 55, 117. (5) Tomasi, J.; Bonaccorsi, R.; Cammi, R.; Olivares del Valle, F. J. J. Mol. Struct. (Theochem) 1991, 234, 401. (6) Cramer, C. J.; Truhlar, D. G. In Reviews in Computational Chemistry; Κ. B. Lipkowitz and D. B. Boyd, Eds.; VCH: New York, 1994; Vol. 6; in press. (7) Onsager, L. Chem. Rev. 1933, 13, 73. (8) Kirkwood, J. G. J. Chem. Phys. 1934, 2, 351. (9) Onsager, L. J. Am. Chem. Soc. 1936, 58, 1486. (10) Reynolds, J. Α.; Gilbert, D. B.; Tanford, C. Proc. Natl. Acad. Sci., USA 1974, 71, 2925. (11) Hermann, R. B. Proc. Natl. Acad. Sci., USA 1977, 74, 4144. (12) Ooi, T.; Oobatake, M.; Nemethy, G.; Scheraga, H. A. Proc. Natl. Acad. Sci., USA 1987, 84, 3086. (13) Cramer, C. J.; Truhlar, D. G. J. Comput.-Aid.Mol.Des. 1992, 6, 629. (14) Warwicker, J.; Watson, H. C. J. Mol. Biol. 1982, 174, 527. (15) Bashford, D.; Karplus, M.; Canters, G. W. J. Mol. Biol. 1988, 203, 507. (16) Davis, M. E.; McCammon, J. A. Chem. Rev. 1990, 90, 509. (17) Honig, B.; Sharp, K.; Yang, A.-S. J. Phys. Chem 1993, 97, 1101. (18) Momany, F. Α.; McGuire, R. F.; Burgess, A. W.; Scheraga, H. A. J. Phys. Chem. 1975, 79, 2361. (19) Lifson, S.; Hagler, A. T.; Dauber, P. J. Am.Chem.Soc. 1979, 101, 5111.

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An Overview

Berendsen, H. J.C.;Potsma, J. P.M.;van Gunsteren, W. F.; Hermans, J. In IntermolecularForces;B. Pullman, Ed.; Reidel: Dordrecht, 1981. Brooks, C. R.; Bruccoleri, R. E.; Olafson, B. D.; States, D. J.; Swaminathan, S.; Karplus, M. J. Comp. Chem. 1983, 4, 187. Clementi, E. In Structure and Dynamics: Nucleic Acids and Proteins; E. Clementi and R. H. Sarma, Eds.; Adenine Press: New York, 1983; p. 321. Weiner, S. J.; Kollman, P. Α.; Nguyen, D. T.; Case, D. A. J. Comp. Chem. 1986, 7, 230. Jorgensen, W. L.; Tirado-Rives, J. J. Am. Chem. Soc. 1988, 110, 1657. Brooks, C. L.; Karplus, M.; Pettit, Β. M. Adv. Chem. Phys. 1989, 71, 1. Warshel, A. Computer Modeling of Chemical Reactions in Enzymes and Solutions; Wiley-Interscience: New York, 1991. Åqvist, J.; Warshel, A. Chem. Rev. 1993, 93, 2418. Whitnell, R. M.; Wilson, K. R. In Reviews in Computational Chemistry; K. B. Lipkowitz and D. B. Boyd, Eds.; VCH: New York, 1993; Vol. 4; p. 67. Jorgensen, W. L. Acc. Chem. Res. 1989, 22, 184. Marcus, R. A. Annu. Rev. Phys. Chem. 1964, 15, 155. Karelson, M. M.; Zerner, M. C. J. Phys. Chem. 1992, 96, 6949. Lee, S.; Hynes, J. T. J. Chem. Phys. 1988, 88, 6863. Hanggi, P.; Talkner, P.; Borkovec, M. Rev. Mod. Phys. 1990, 62, 251. Truhlar, D. G.; Schenter, G. K.; Garrett, B. C. J. Chem. Phys. 1993, 98, 5756. Basilevsky, M. V.; Chudinov, G. E.; Newton, M. D. Chem. Phys. 1994, 179, 263. Truhlar, D. G.; Garrett, B. G. Annu. Rev. Phys. Chem. 1984, 35, 159. Grieco, P. A. Aldrichim. Acta 1991, 24, 59. Li, C.-J. Chem. Rev. 1993, 93, 2023. Warshel, Α.; Levitt, M. J. Mol. Biol. 1976, 103, 227. Jensen, J. H.; Day, P. N.; Gordon, M. S.; Basch, H.; Cohen, D.; Garmer, D. R.; Krauss, M.; Stevens, W. J. In Modelling the Hydrogen Bond; D. A. Smith, Ed.; American Chemical Society Symposium Series: Washington, DC, 1994; in press. Field, M. J.; Bash, P. Α.; Karplus, M. J. Comp. Chem. 1990, 11, 700. Gao, J. J. Phys.Chem.1992,96, 537. Stanton, R. V.; Hartsough, D. S.; Merz, K. M. J. Phys. Chem. 1993, 97, 11868. Ten-no, S.; Hirata, F.; Kato, S. Chem. Phys. Lett. 1993, 214, 391. Warshel, Α.; Åqvist, J.; Creighton, S. Proc.Natl.Acad. Sci., USA 1989, 86, 5820.

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Hermann, R. B. J. Phys.Chem.1972, 76, 2754. Huque, E. M. J. Chem. Ed. 1989, 66, 581. Blokzijl, W.; Engberts, J. B. F. N. Angew.Chem.,Int. Ed. Engl. 1993, 32, 1545. Nemethy, G.; Scheraga, H. A. J. Chem. Phys. 1962, 36, 3401. Kellis, J. T.; Nyberg, K.; Sali, D.; Fersht, A. R. Nature 1988, 788,

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Chapter 2

Application of Continuum Solvation Models Based on a Quantum Mechanical Hamiltonian

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J. Tomasi Department of Chemistry and Industrial Chemistry, University of Pisa, Via Risorgimento 35, 56126 Pisa, Italy

This chapter discusses areas of study of chemistry in solution for which continuum models may be used profitably. A distinction among types of projects is introduced. The projects may be computational applications, in which existing computer codes are used to get numerical values of a desired property, or they may be methodological studies, addressed to the implementation of further elaboration of the basic procedure. Solvation energy, reaction mechanisms, energy derivatives, and local and large scale anisotropies are the main topics considered here. The exposition is mainly based on the past experience of our group, with the inclusion of some recent developments.

The main characteristic of continuum models of solvation is their simplicity in the description of the solvent structure. This quality is quite appealing for several reasons, in particular the possibility of extending such models to treat a large number of processes and systems, the ease of interpretation of experimental and computational results, and efficiency in routine calculations. The simplicity is tempered in quantum mechanical versions with a more accurate representation of the solute by means of ab initio or semi-empirical methods. We may draw from these general statements some indications about the most promising ways of using continuum models. For this purpose we shall consider some specific topics where modelling and elaboration of efficient computer codes have different importance: the evaluation of the solvation energy, the description and interpretation of chemical reactions, and the elaboration of more detailed continuum descriptions of the solvent. These examples, to which more could be added, also show when and how quantum continuum methods are useful. This choice has been suggested by our personal experience, and we

0097-6156/94/0568-0010$08.00/0 © 1994 American Chemical Society

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

2. TOMASI

Use of Continuum Solvation Models

11

will mainly rely on the use of the polarized continuum model (PCM), a computational procedure we proposed years ago (1) and are continuously refining, as some previews in the following pages will show.

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Models and routine computational codes Modelling of solvent effects via continuous models dates back to the beginning of the molecular description of solutions and continues into current years. Einstein (2) in 1906 used a continuous model to describe transport properties of solutes (continuous models are not limited to electrostatic effects); the fundamentals of the theory of ionic solutions are given by the 1920 continuum model of Born (3); the basic aspects of electron transfer reactions are described by the 1956 theory of Marcus (4) invoking solvent fluctuations in a continuum model; finer aspects of similar reactions may be interpreted using a quantum description of the fast component of the continuum electrostatic polarization, as Gehlen et al. suggested in 1992 (5). These few examples, to which many others could be added, show that the persistence in the use of continuum models is accompanied by a remarkable versatility of the basic concept that surely will continue being exploited in the future. The efficiency of some recent semi-empirical methods in routine computation of solvation energies of medium size molecules (6) is well documented and accompanied by a satisfactory quality of the results. The same methods may be used to describe reactions in solution: the thermodynamic balance of a reaction may be well reproduced (6), but description and interpretation of the reaction mechanism is limited by the approximations of the semi-empirical approach. Similar considerations hold for semi-classical computational codes. These codes are not based on quantum calculations in solution, but they may rely on quantum calculations in vacuo to get the necessary parameters. Extended Born methods yeld fairly accurate solvation energies (7). Further extensions and refinements of the semi-classical codes will profit from analyses of continuum quantum results. T h e Role of ab initio C o n t i n u u m Solvation Methods. We shall consider ab initio continuum methods as the primary object of this paper. There are two reasons for this choice. The first has a methodological character. A sound strategy to test qualities, defects, and potentialities of a model consists in examining the output of the model at its best, in the most complete and detailed form, and then reducing it to a more manageable level when the essential features to be preserved are well ascertained. Shortcuts are not advisable; they may lead to wrong conclusions. The theoretical chemistry literature is rich in wrong statements based on a hurried examination of a model; some of these statements refer to continuum solvation models. When the usual homogeneous continuum model is considered, the description of the solute charge distribution and of the mutual solute-solvent interaction effects should be done at the highest possible level of accuracy to detect limits and potentialities of the approach. Analogous considerations hold when the homogeneous medium is replaced by other continuum distributions.

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

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STRUCTURE AND REACTIVITY IN AQUEOUS SOLUTION

The second reason is related to the intrinsic superiority of ab initio quantum methods in describing fine details of molecular charge distributions. For some problems a detailed ab initio description is absolutely necessary. The Solvation Free Energy One of the basic problems in the study of solutions is the development of models able to describe their thermodynamic properties, for example the free energy of solvation, A G i , of neutral solutes at infinite diluition. From the application of ab initio continuum methods we may state the following points: 1) Calculations of A G i (and of related quantities) with continuum models are able to yield results within the range of experimental errors. 2) These calculations require the use of a well shaped cavity. The use of cavities with simple shapes can be accepted only under limited conditions. 3) Electrostatic (Coulomb and polarization), dispersion, cavitation, and repulsion terms are all necessary: A G i = AG i+Gdis+Gcav+G epThe relative importance of these terms may be related to the bulk properties of the solvent and to the molecular properties of the solute. In making comparisons among solutes of the same class some terms may be neglected, but this choice must be based on the information derived from the decomposition of complete ab initio results. 4) The extra energy effects due to local variations induced in the solvent distribution by the solute (cybotactic changes) are of limited entity. 5) The contributions due to terms related to the vibrational, rotational and translational solute partition functions are not decisive for almost rigid solutes. Some corrections due to large-amplitude motions (hindered internal rotations, out-of-plane deformations) and to zero point contributions (stretching of M-H groups making hydrogen bonds with solvent molecules) may be easily introduced, when necessary. The semi-empirical procedures we have mentioned satisfy points 2, 3 and 4. The analysis of ab initio results justifies their formulation and suggests some minor improvements.

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so

s o

8 0

e

r

N u m e r i c a l Results. We report here the results of a linear regression analysis of computed against experimental AGhyd values. The computed values refer to the P C M ab initio and to the AMSOL-SM2 semi-empirical procedures. The data reported in the Table are the coefficients of the regression line: AGhyd(exp) = a AGhyd(comp) + b (Kcal/mol), the regression coefficient R, the standard error σ and the number η of cases within each set. Set 1 refers to a sample of chemically similar compounds (esters) all described at the 6-31G* SCF level with geometry optimisation in water (8). Set 2, presented here for the first time, it is not chemically homogeneous (neutral organic solutes with heteroatoms) and is computed using the same 6-31G* basis set with geometries optimised in vacuo. Some of the

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

2. TOMASI

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Use of Continuum Solvation Models 1

"experimental' values are drawn from the empirical formulas elaborated by Cabani et al.(9). Set 3 includes all the neutral polar solutes used by Cramer and Truhlar in their calibration procedure (β), but not the charged ones. The last set has been also computed by Cramer and Truhlar, but it was not used in the calibration. These results show that ab initio calculations may reach chemical accuracy and that semiempirical values are of a good level.

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Table I. Model Predictions against Experimental Data Set

Method

a

b

R

σ

η

1 2 3 4

PCM PCM AMSOL AMSOL

1.03 0.98 0.90 0.98

0.33 0.35 -0.02 0.62

0.999 0.910 0.876 0.965

0.015 0.452 1.207 0.796

16 102 117 7

Good results are also obtained with other semiempirical procedures. The results are somewhat scattered and we refer to a still unpublished review by Cramer and Truhlar (Cramer C.J.; Truhlar, D.G. Reviews in Computational Chemistry 1994, in press) for more information. Note, however, that this field is in rapid evolution and that the number of methods and the quality of the results are increasing rapidly. The reduction of the model may be carried further. Even semiclassical models give results of appreciable quality (see again the review by Cramer and Truhlar quoted above). A sequence of approximations starting from the quantum description and ending with atomic charges may be a guide to check this reduction of the model without shortcuts (10). The continuum methods compare well with other approaches. The first systematic comparison between ab initio continuum and M D based free energy perturbation calculations of AGhyd performed by Orozco, Jorgensen and Luque (11) gives an average error of 0.8 kcal/mol for the continuum P C M procedure and 1.5 kcal/mol for the M D - F E P technique with respect to the experimental values. Of course many refinements may be introduced in this picture, but details are not essential here: we conclude that computationally very convenient continuum methods may be used to obtain gas-liquid and liquid-liquid transfer thermodynamical properties. Chemical Reactions with the Continuum Model The evaluation of solvation energy is but a tiny part of the topics facing theoretical chemistry in solution. We shall consider now a more challenging subject, the description of chemical reactions. Here the ab initio formulation of the continuum model has an important role. Let us summarise the formal set-up of the approach. Continuum methods are able to give an evaluation of the free energy G of the solute in In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

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STRUCTURE AND REACTIVITY IN AQUEOUS SOLUTION

solution (not to be confused with A G i ) as a function of the nuclear coordinates {R) of the solute ( as "solute" we consider the bare reacting molecules, supplemented when necessary with a restricted number of solvent molecules playing an active role in the reaction). The G(R) surface corresponds to the E(R) surface defined in vacuo. We apply to the G(R) surface (or, better, to the set of relevant G(R) surfaces) the same concepts originally derived for reactions in vacuo. First, we define geometry, energy and electronic structure of reactants, products, intermediates, and saddle points. Then, we define the reaction coordinate and the portions of the energy surface near the reaction path necessary for dynamical studies of the reaction. The chemical interpretation of the mechanism will be based on a scrutiny of the solute wave function, to be performed with suitable techniques. This simplified version of a rather formidable problem (a quantum study of a system composed of a large number of molecules) is corroborated by preliminary tests on the model. In particular the partition of the whole system into a "solute" and a medium is supported by the success in describing the energy profile of several significant reactions. This scheme must be supplemented when certain dynamical effects of the solvent are considered. It is known that in many important classes of reactions the dynamics cannot be properly described by relying on the G(R) surface alone: the typical case is the outer-sphere electron transfer model of Marcus (4), in which the dynamics is carried by a solvent coordinate alone, without intervention of the geometric coordinates of the solute. We are thus compelled to extend our definition of energy hypersurface G(R) by including some extra dynamical coordinates {S}, and to use, when and where necessary, a more general function G(R+S). This enlargement of the space is not equivalent to the addition of more solvent molecules in the "solute". We have thus recognised at least three important problems: the correct evaluation of G(R) and of its critical points, the description of the electronic structure at some significant points, and the definition of the additional {S} subspace, supplemented by the protocols for the use of G(R+S)

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so

The Analysis of the G(R) Surface. Long experience in the evaluation of E(R) surfaces tells us that semiempirical methods give, in the most favourable cases, nothing more than a first order guess. Ab initio calculations, of good quality, are necessary. The analysis of the G(R) surface is made easier by the use of the gradient of G(R), grad G(R), supplemented by the diagonalization of the Hessian matrix H(R). The calculation of grad G and of H must be performed according to the conditions set in points 2 and 3 above, i.e. using a suitably shaped cavity and including in G(R) all the necessary contributions: G i(R) + Gdi (R) + G ( R ) + G ( R ) . Current ab initio continuum programs are not equipped for the analytical evaluation of gradG and H at this level of accuracy. We present here the computational scheme we have recently elaborated in the framework of the P C M (Cammi, R.; Tomasi, J., J. Chem. Phys., May 1994). e

s

c a v

rep

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

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Use of Continuum Solvation Models

Iterative a n d Direct P C Method. The original iterative P C M version is not suited to get analytical derivatives of G i (the other terms of G are simpler to manage and may be treated separately). It is more convenient to resort to a matrix formulation of the electrostatic problem, exploiting the fact that in the PCM the solute is effectively replaced by a charge distribution σ on the cavity surface, and that this surface is divided into a finite number of tesserae. An accurate elaboration of the matrixP C M has been done by the Sakurai group (12). We have elaborated a similar procedure, more computationally effective and suitable to compute the first and second derivatives of G i with respect to parameters α and a, β, later indicated by G and G P respectively. This preliminary step highlights some points that deserve mention. In the iterative procedure the Schrodinger equation e

e

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a

A

|(Η0 + ν ) Ψ > = Ε | Ψ >

(1)

σ

may be solved with the traditional variational techniques because V, and Vq to which nuclear contributions must be added (1):

G = - 1 / 2 < Ψ | ν | Ψ > + ν Ν Ν + 1 / 2 υ . σ

σ

Ν σ

(2)

Here V N N is the intrasolute nuclear repulsion term also present in vacuo calculations and U N O is the interaction between the solute nuclei and the apparent surface charge σ. In the direct methods simultaneously optimizing the solute and solvent charge densities, the functional to be minimized is not the mean value of H° + ν , but rather the free energy fuctional G (13). When the problem is recast at the Hartree-Fock level, with expansion over a finite basis, the minimization of G is reduced to the solution of a pseudo-HF equation (12): σ

f

F C = ESC

(3)

with F» = h» +G'(P) = (h + 1/2(J + Y)) + (CKP) + X ( P » .

(4)

Here h and G(P) are the usual H F one-electron and two-electron integrals matrices, Ρ is the density matrix, J , Y and X(P) collect one-and twoelectron integrals involving interactions with the surface apparent charges. (To be more precise, J describes the interactions between the elementary solute electron charge distributions and the surface charges having as origin the solute nuclear charges, Y describes the interactions between the solute nuclei and the surface charges having as source the elementary In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

16

STRUCTURE AND REACTIVITY IN AQUEOUS SOLUTION

solute electronic distributions, X describes the interactions between elementary electron distributions and the surface charges they generate). The expression of F is different from that of the Fock matrix related to equation (1) and applied in the iterative procedure. The two approaches are formally equivalent (Cammi, R.; Tomasi, J . J. Comp. Chem., to be published) and must give the same values for G and for the coefficients C of the wave function.

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1

The Renormalization of the Surface Charges. This formal equivalence is numerically verified only when a renormalization of the surface charge σ (or of the point charges q which describe σ) is introduced in the computational scheme. The integrated value of σ must in fact satisfy a simple relationship with the total charge Q M of the solute:

Jo(s)ds =(ε-ΐνε Q .

(5)

M

Similar relationships hold for the electronic and nuclear components of the surface charge (σ = o + σ ) having as sources the solute electron and nuclear charge distributions ( Q M = Q M + Q M )· These conditions are not satisfied by continuum quantum calculations (one reason is that a portion of the electronic charge distribution is spread out of the cavity, by definition). The effect of this lack of normalization is different in the iterative and direct methods and results without renormalization may differ. On this basis it has been stated that there are two alternative pictures for the description of the solvation energy. Actually, there is only one picture, and the choice is between two alternative but equivalent computational methods. The introduction of a suitable renormalization permits one to recover in actual computations the equivalence of the two methods. In addition, the renormalization permits to exploit the formal equivalence between J and Y (the first, as said, describes the interactions of the elementary electronic solute charge distributions with σ , the second describes the interactions of nuclei with 0 ) that is lost in un-normalized calculations, with a further reduction of the computational times. With this reformulation the ab initio direct method is almost as fast as the iterative method for solutes of small size. e

Ν

e

N

Ν

e

A n a l y t i c a l Derivatives w i t h the P C Method. Surface charge renormalization plays an even more important role in the analytical determination of derivatives. Without renormalization no meaningful analytical derivatives may be computed in the ab initio apparent surface charge methods. The expression of G , obtained using equation (3), is: 1

f

G = trPh + 1/2 trPG (P) + V N N

(6) N

where V ' N N = V N N + 1/2 U N N collects nuclear repulsions and nuclei-o contributions to G . From this definition of G we may derive formal expressions for its first and second derivatives with respect to the cartesian components α and β of the solute nuclear coordinates: In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

2. TOMASI

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Use of Continuum Solvation Models f

G« = trPh*« + 1/2 trPG a(P) - trS e

m

h

is the density of states, or spectral density. Fig. 7 collects the results in the frequency range 0 - 4 5 0 0 c m at Τ = 305K for liquid H 0 (left), which shows frequency shifts relative to both liquid D 0 (right) and to the gas phase. The gas phase frequencies are indicated with dotted lines in the two insets. In the figure, the oxygen atom contribution is separated from that of the hydrogen (or deuterium) atom. Moving from low to high frequency, the main bands have been assigned, on the base of laboratory experiments, to the translational, librational, bending, and stretching modes, respectively. - 1

2

2

- 0 ii

-D _. gas

ri : i Π

500.

1500.

2500.

3500.

4500.

: ! !

500.

1500.

!

2500.

!

cd(cm "') 3500.

4500.

Figure 7. Left: Oxygen and hydrogen power spectrum from simulated H 0 at Τ = 305 Κ. Right: Same but for D 0 . 2

2

Notice that at low frequencies the spectrum is dominated by the contribution from the oxygen atom trajectories, while at high frequencies the hydrogen atoms domi­ nate with the deuterium at intermediate frequencies (this is expected on the base of the H/D relative masses). The low frequency region is dominated by the intermolecular water-water interactions, while the high frequency region is dominated by the intramolecular motions for a single molecule of water. In the following, we shall first provide an overall view of the density of state in the liquid, up to 4500 cm-1, and then we shall comment band by band in detail. The gas phase values of Fig. 7 have been computed with normal mode analysis. By comparing the computed gas phase values with the corresponding experIn Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

7.

CLEMENTI AND CORONGIU

103

Simulated Water Structure

imental values (56, 57), it is found that the three vibrational frequencies (the bending, v , and the stretching modes symmetric, V i , and antisymmetric, v ) are overestimated by 90, 189 and 199 cm-1, respectively for H 0 and by 55, 110, and 113 cm-1 f D 0 (56). These differences arise because the experimental values account for both the harmonic and anharmonic contributions to the vibrations, whereas the normal mode analysis is limited to the harmonic part only. It is note­ worthy in this respect to recall that experimental values (57) have estimated the harmonic contribution for H 0 to be 1649, 3832, 3943, cm-1, for v , V! and v respectively not far from the computed values of 1685, 3846, 3955 cm- . In Fig. 7, left inset, the band centered at ~ 1750 c m is the intramolecular bending mode, while those at 3626 and 3694 cm- are the intramolecular O H stretching, sym­ metric and asymmetric, respectively. Comparing these frequencies with those of the single molecule gas phase (vertical dotted lines), we obtain a - 6 5 cm" up shift for v and down shifts of -220 and -261 cm- for Vi and v respectively. These shifts should be compared with the experimental values (38-40) of 55 cm-1 for v , and ~ -300 cm-1 for the stretching modes (experimentally Vi and v are not well resolved). Therefore, from the liquid simulation we obtain bending and stretching bands which substantially deviate from the free molecule and show characteristics typical of the liquid state. This is an indication that the anharmonicity of the intermolecular //-bond interactions are reliably represented by our model. The bending mode, found in the liquid phase at 1650 cm- in Ref. 39, should be compared with our value of ~ 1750 c m for 7 = 305 K . In our simulations the position of the bending mode is overestimated by about 100 cm- , and the stretching by about 250 cm- . These overestimations are about as large as the anharmonic effects discussed above for the gas phase. The librational, bending and stretching D 0 spectra in the right inset of Fig. 7, are essentially very similar, in appearance, with those of H 0 except that for D 0 the bands are narrower and more pronounced (see, for example, the very good separation between Vj and v , with a split of about 90 cm- ). Concerning the temperature dependency of the translational, librational, bending and stretching modes for both H 0 and D 0 it has been observed that only the bending mode is essentially temperature independent, for the other modes, shifts are obtained from our simulations. For details see Ref. 25. 2

3

2

or

2

2

2

3

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1

-1

1

1

1

2

3

2

3

1

-1

1

1

2

2

2

1

3

2

2

A model for liquid water Since the pioneering works of Rontgen (41) and Bernai and Fowler (42) there have been many attempts to explain the unusual properties of water in terms of simple models, which can be subdivided grosso modo in two broad categories: The "continuum" and the "mixture" models. The "continuum model" assumes that the dominant structure of the liquid is a "locally tetrahedral continuous hydrogen-bonded network", (43, 44) whereas the "mixture models" propose (45) the existence of a thermodynamic equilibrium between two or more different aggregates of water molecules. Among the recent proposed "mixture models", we recall those presented in Refs. 46-50. We recall in addition that a detailed analysis of Raman spectra by Walrafen et al. (39) sup­ ports the hypothesis of two components: the HB and the NHB, corresponding to four-hydrogen bonded and 3-hydrogen bonded water molecules, respectively.

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

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STRUCTURE AND REACTIVITY IN AQUEOUS SOLUTION

Our M D simulations support the hypothesis of an equilibrium between water molecules hydrogen bonded to two, three, four and five water molecules, in agreement with previous proposal (23, 47); in particular, we present an analysis, where we predict specific spectral features for the different species present in liquid water. Analysis carried out on the stored trajectories revealed (23) that liquid water can be viewed as composed of "clusters" of different sizes and with different probabilities of being present in the liquid, depending also from the temperature. We recall that a "cluster" can be characterized either by its oxygen atom positions or by the position of both the oxygen and the hydrogen atoms; we have placed quotation marks on the term cluster, to emphasize that a cluster in the liquid is different from a cluster in vacuo. Indeed, the former relates to bulk water, the latter to surface structures. A cluster in the liquid is defined as an association of (n+1) water molecules, i.e. a central water molecule coordinated, or hydrogen bonded (HB), to n; alternatively, if we refer specifically to the central molecule, we talk of a bi-, three-, η-coordinated water molecule (n=2, n=3, etc.). In Fig. 8 we present four clusters. The label A , B, C, and D identifies the central molecule in each cluster, i.e. the "solvated water molecule". The solvated molecules A and Β are examples of tetra-coordination. Notice that A is coordinated to four water molecules (ice-like structure) via four hydrogen bridges (two between the lone pair electrons of A and hydrogen atoms of two solvating water molecules and two between the hydrogen atoms of A and the oxygen atoms of two additional solvating water molecules). In Β the situation is different, since we have one hydrogen atom of the solvated molecule Β bridging two oxygen atoms of two solvated water molecules (bifurcated hydrogen bond); two additional water mole­ cules are bridging the lone pairs of B, as for A . Molecules C and D are examples of three-coordinated water molecules: C has a free lone pair, D has a free hydrogen atom.

Β

C

Figure 8. A and B are examples of tetra-coordinated water molecules. C and D are examples of water molecules three-coordinated. Reprinted with permission from Ref. 51. Copyright 1993 Elsevier Science Publishers B.V. An analysis (23) carried out by considering only oxygen atoms, revealed that at low temperature n=4 is more frequent than n=5, whereas n=5 is more probable at high temperature (see Fig. 9 left insert).

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

7. CLEMENTI AND CORONGIU

105

Simulated Water Structure 1.0

l.Oi n /n e

w

- n /n

R=R**,(goo)

e

0.8h

0.8 h

0.6r-

0.6

R=R *,(9OK)

w

w

:

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*^*4

0.4-

0.4

^4 0.2H

*6

0.2

7 : 0.0 m Îi^wi4itTîrirrCC^rrl"iTi"i111t11 iTi ι 0.0 m i m-rrrhmf Τίί ιΤι1111111111 240 280 280 320 ^ j 360 240 Τ(·Κ) T

3 2 0

K

3

6

0

Figure 9. Distribution of water molecules with coordination number from 2 to 7. Left: O-O coordination number within a sphere with R = R j of gooRight: O H coordination number within a sphere with R = R j of gQH* m

m

n

n

Three-, hexa-and epta-coordination exist, but less abundantly. Instead, by consid­ ering both the oxygen and hydrogen atoms, then the analysis revealed that, at all the temperatures, the tetra-coordination is the most important, with the three- and penta-coordinations becoming more and more abundant as the temperature rises (see Fig. the right inset of Fig. 9). The lifetime of the tetra-coordinated water molecules is the longest; from the analysis we found in addition smaller and larger clusters, but the corresponding populations are small and the lifetime vanishingly short. The resulting overall picture of liquid water is that of a very dynamical "macromolecular" system, where clusters of different size and structure coexist in different subvolumes of the liquid and each has characteristic lifetimes and specific temperature dependencies. For the liquid at T=3()5 K, we report the density of states of the hydrogen atoms belonging to the water molecules bi-, three, tetra- and penta-coordinated, defining the coordination number by considering both oxygen and hydrogen atoms. Thus, we do not analyze the spectrum in terms of the entire liquid, but we generate the spectra corresponding to different types of solvated water molecules, each type with its specific coordination. In Fig. 10 we report the density of states in the region 0-4500 cm-1 obtained by considering the hydrogen atoms belonging to water molecules with n=2, 3, 4, and 5. The first band extends up to 1000 cm-1 j corresponds to the librational motions, the second one to the bending mode (v ), and the last one to the stretching modes, symmetric (Vi) and asymmetric (v ). The dotted vertical lines refer to frequencies for one single water molecule in vacuo, computed with the same potential we have used to study the liquid. We have labelled as "total" the previously reported (see the left inset of Fig. 7) spectrum obtained by considering the entire sample of the liquid without any subdivision into clusters. In this spec­ trum the modes v , v and v occur at ~ 1756, ~ 3626, ~ 3694 cm-1, respectively. The spectra labelled as "2", "3", etc. refer to water molecules with n=2, n=3, etc. a n c

2

3

2

u

3

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

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106

STRUCTURE AND REACTIVITY IN AQUEOUS SOLUTION

1000.

2000.

3000.

4000.

1000.

2000.

w(cm "')

3000.

4000.

u(cm "')

Figure 10. Left: Density of states of the hydrogen atoms. "Total": full liquid water sample; "2": only bi-coordinated water molecules; "3": only threecoordinated water molecules; "4": only tetra-coordinated water molecules; "5": only penta- coordinated water molecules. Right: "Total": full liquid water sample; "0": hydrogen atoms not hydrogen bonded to any oxygen atom; "1": hydrogen atoms hydrogen bonded to one oxygen atom; "2": hydrogen atoms hydrogen. Adapted from Ref. 51. In general, by comparing this set of data, notable differentiating features can be noted at all frequencies. In particular, in the librational region, the water mole­ cules bi-coordinated show a maximum at -380 cm-1, which blue shifts at -450 cm-1 for the three-coordinated and to -510 cm-1 for the tetra-coordinated. Fol­ lowing this first maximum a shoulder starts to appear at high frequencies for the three-coordinated water molecules, and it becomes more evident for the tetracoordination. The shoulder developes into a broad maximum, centered at -650 cm-1 (extending up to 1000 cm-1 for the penta-coordinated water molecules). The superposition of these partial bands yields the curve reported as "total". Walrafen et al. (39) assuming for the tetra-coordinated water molecules a C symmetry as in ice (i.e. four nearest neighbors with four hydrogen bonds equal in length, angle, etc., two for its two protons and two lone pairs) report the libration around the C axis of A symmetry at 550 cm-1, the in-plane libration of B sym­ metry at 425-450 cm-1, d the out-of-plane libration of B symmetry at 720-740 cm-l. From Fig. 10 we notice for the terra- coordinated water molecules a maximum at -510 cm-1 d a shoulder at - 730 cm-1; it should be noted that one could assume under the first broad maximum another gaussian distribution at lower frequencies. We have already pointed out that in our analysis tetra-coordination does not necessarily implies one water molecule with four neighbors as in ice (as for A in Fig. 8). In our definition tetra-coordination can arise from many and different con­ figurations: for example, either the one considered above (however with a distribution of HB lengths and angles), or one with the lone pairs coordinated to two water molecules and one hydrogen atom hydrogen bonded to two different oxygen atoms (bifurcated hydrogen bond) and the second hydrogen atom free, etc. It is reasonable to assume that the first type (the ice-like structure) is the most prob2 v

2

2

a n

2

x

a n

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

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7. CLEMENTI AND CORONGIU

107

Simulated Water Structure

able. These possible combinations can coexist for all the distributions of coordi­ nations given in Fig. 9, preserving, however, the total number of HBs for each combination. In Fig. 8 water molecules C and D provide examples for two dif­ ferent combinations of three-coordination. Comparing the spectra corresponding to different coordinations in the bending region, we observe small variations in frequency; for example the pentacoordinated water molecules show a maximum at -1780 cm-1, which progres­ sively shifts towards lower frequencies (towards the gas phase value) as the coordination number decreases reaching a value of -1735 cm-1 for the bicoordination. More evident are the shifts in the stretching region. Starting from the pentacoordination and decreasing the coordination number, we notice a clear shift towards higher frequencies (moving towards the gas frequencies) pointing out the presence of an increasing percentage of water molecules with one hydrogen atom unbounded. Notice also that for all the coordinations, except for n=5, we find very well distinct peaks for the symmetric and asymmetric stretching modes. To provide additional details on the role played by the "free" hydrogen atoms of the solvated water molecules, we have attempted the following analysis. We have subdivided the water molecules in the liquid into different distributions, each one being characterized by the number (0, 1, 2) of hydrogen bonds for a given hydrogen atom. In particular, when for one water molecule one hydrogen atom is free (i.e. it has not HB to any molecule) it belongs to group "0", when it is HB to a water molecule (i.e. pointing towards the oxygen lone pair), it belongs to group "1", when HB to two water molecules it belongs to group "2", independently from the second hydrogen atom, for which the same kind of analysis is carried out. In Fig. 8, examples for 0, 1, and 2 are, respectively, the hydrogen atom 2 of mole­ cule D, the hydrogen atom 2 of molecule B, and the hydrogen atom 1 of molecule B. By averaging over all the hydrogen atoms belonging to a given group and over all the simulation time steps, we obtain the data reported in the right inset of Fig. 10. Again, the data labelled as "total" refers to the entire liquid sample. Those hydrogen atoms with no HB show in the librational region a maximum at -350 cm-1, in the bending region a maximum at -1735 cm-1 and in the stretching region a peak at -3790 cm-1. Those hydrogen atoms with one HB (the most probable type) show in the librational region a maximum at -530 cm-1 and a shoulder, which covers a gaussian with a maximum at -730 cm-1. \ the bending region the maximum occurs at -1770 cm-1 and finally, in the stretching region the symmetric and asymmetric modes occur at -3630 cm-1 and -3690 cm-1, respec­ tively. The hydrogen atoms with two HBs (bifurcated hydrogen bond) show in the librational region a broad maximum centered at -550 cm-1; the bending mode occurs at -1760 cm-1 and in the stretching region a non-split maximum is present at -3670 cm-1. Note that the population, at T=3()5 K, of the 0, 1, and 2 distrib­ utions are vastly different, i.e. large for 1, very small for 2 and small for 0. n

Conclusions The NCC-vib interaction potential has been obtained by fitting ab initio data rather than experimental data. This potential represents the latest stage of an evo­ lution, which started with a Hartree-Fock potential (14), after several systematic refinements (75-/9), has brought to the M C Y model. As one can expect, this evo­ lution takes advantage both of the increased performance of computational means In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

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108

STRUCTURE AND REACTIVITY IN AQUEOUS SOLUTION

and of advances of theoretical nature. The present potential has accurately reproduced much structural and dynamical data, predicted overall trends and very detailed features observed in infrared, Raman and neutron scattering experiments, both for the liquid and the solid phases. The M D trajectories have been used as "experimental data" to develop a descriptive model for liquid water, which is much richer of quantitative features than simply assuming essentially a network of tetrahedrally oriented water molecules. We have proposed the model of a dynam­ ical liquid, which can be represented as a temperature dependent mixture of "clusters" of different size and lifetime, with well defined probability. Individual water molecules break off from one cluster to build up another cluster. The hydrogen bond is energetically affected by the temperature, as we have reported (see Ref. 23) and by its belonging to a given type of cluster. Its 0 - 0 distance and O-H-0 angle varies with temperature. There is a complex distribution of life­ times for the hydrogen bond in liquid water, even for a given temperature, since it depends also on the cluster to which it belongs. Typically we can say that a hydrogen bond in liquid water can last up to a few picoseconds. Work is in progress to elucidate further aspects of liquid water using a recently completed molecular dynamics code, where the forces are obtained at each time step using Density Functional Theory and Gaussian basis sets (51). The computations presented in this work are very computer intensive tasks, both in the generation of the N C C potential and in performing the molecular dynamics simulations and corresponding analysis. We recall that quantum chem­ istry (needed for constructing the ab initio potentials) and molecular dynamics simulations are areas particularly well suited to parallel computers. In this context we recall that computational chemists use more and more from clusters of work­ stations to parallel systems like the IBM-SP1, the Crsy T3D, or the CM5 of the Thinking Machine Corporation. It is worth noting that the European Communities Commission has established a special program to parallelise scientific codes, key to the industry, under the ESPRIT-ΠΙ Programme. In the area of computational chemistry there are two projects, coordinated by Smith System Engineering Limited, U K , whose task is the porting to parallel platforms of computer codes both in quantum chemistry and molecular dynamics modelling. Acknowledgement It is a pleasure to acknowledge financial support from the "Regione Autonoma della Sardegna". References 1. Rahman, Α.; Stillinger, F.H. J. Chem. Phys. 1971, 55, 157. 2. Lemberg, H.L.; Stillinger, F.H. J. Chem. Phys. 1975, 62, 1667. 3. Stillinger, F.H.; Rahman, A. J. Chem. Phys. 1978, 68, 666. 4. Watts, R.O. Chem. Phys. 1977, 26, 367. 5. Reimers,J.R.;Watts, R.O. Mol. Phys. 1984, 52, 357. 6. Coker, D.F.; Watts, R.O. J. Phys. Chem. 1987, 91, 2513. 7. Barnes, P.; Finney,J.L.;Nicholas,J.D.;Quinn, J.D. Nature 1979, 282, 459. 8. Jorgensen, W.L. J. Am. Chem. Soc. 1981, 103, 335. 9. Jorgensen, W.L. J. Chem. Phys. 1982, 77, 4156. 10. Berendsen, H.J.C.; Postma, J.P.M.; van Gunsteren, W.F.; Hermans,J.Intermolecular Forces, B. Pullman, Ed., Reidel, Dordrecht, (1981). 11. Teleman, O.; Jönsson, B.; Engström, S. Mol. Phys. 1987, 60, 193. 12. Cieplak, P.; Kollman, P.; Lybrand, T. J. Chem. Phys. 1990, 92, 6755. In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

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7. CLEMENTI AND CORONGIU

Simulated Water Structure

109

13. Sprik, M. J. Chem. Phys. 1991, 95, 6762. 14. Popkie, H.; Kistenmaker, H.; Clementi, E. J. Chem. Phys. 1973, 59, 1325. 15. Lie, G.C.; Clementi, E. J. Chem. Phys. 1975, 62, 2195. 16. Matsuoka, O.; Clementi, E.; Yoshimine, M. J. Chem. Phys. 1976, 64, 1351. 17. Carravetta, V.; Clementi, E. J. Chem. Phys. 1984, 81, 2646. 18. Clementi, E.; Corongiu, G. Int.J.Quant. Chem. Symp. 1983, 10, 31. 19. Detrich,J.H.;Corongiu, G.; Clementi, E. Chem. Phys. Letters 1984, 112, 426. 20. Nieser, U.; Corongiu, G.; Clementi, E.; Kneller, G. R.; Bhattacharya, D. J. Phys. Chem. 1990, 94, 7949. 21. Laasonen, K.; Sprok, M.; Parrinello, M. J. Chem. Phys. 1993, 99, 9080. 22. Corongiu, G. Int.J.Quantum Chem. 1992, 44, 1209. 23. Corongiu, G.; Clementi, E. J. Chem. Phys. 1993, 98, 2241. 24. Corongiu, G.; Clementi, E. J. Chem. Phys. 1992, 97, 2030. 25. Corongiu, G.; Clementi, E. J. Chem. Phys. 1993, 98, 4984. 26. Sciortino, F.; Corongiu, G. J. Chem. Phys. 1993, 98, 5694. 27. Sciortino, F.; Corongiu, G. Mol. Phys. 1993, 79, 547. 28. Clementi, E.; Corongiu, G.; Sciortino, F. J. Mol. Struct. 1993, 296, 205. 29. Bartlett, R.; Shavitt, I.; Purvuis, G.D. J. Chem. Phys. 1979, 71, 281. 30. Benedict, W.S.; Gailar, N.; Plyler, E.K. J. Phys. Chem. 1956, 24, 1139. 31. (a) Narten, A.H.; Levy, H.A. J. Chem. Phys. 1971, 55, 2263. (b) Bosio, L.; Chen, S.H.; Teixeira, J. Phys. Rev. A. 1983, 27, 1468. 32. Thiessen, W.E.; Narten, A.H. J. Chem. Phys. 1982, 77, 2656. 33. Soper, A.K.; Phillips, M.G. Chem. Phys. 1986, 107, 47. Soper, A.K.; Silver, R.N. Phys. Rev. Lett. 1982, 49, 471. 34. Narten, A.H. J. Chem. Phys. 1972, 56, 5681. 35. Sciortino, F.; Geiger, Α.; Stanley, H.E. Phys. Rev. Lett. 1990, 65, 3452. 36. Herzberg, G. Molecular Spectra and Molecular StructureII.Infrared and Raman Spectra of Polyatomic Molecules Van Nostrand, Princeton, NJ, 1945. 37. Califano, S. Vibrational States, Wiley, London, 1976. 38. Ratcliffe, C.I.; Irish, D.E. J. Chem. Phys. 1982, 86, 4897. 39. Walrafen, G.E.; Hokmabadi, M.S.; Yang, W.-H. J. Chem. Phys. 1988, 92, 2433. 40. Walrafen, G.E.; Hokmabadi, M.S.; Yang, W.-H. J. Chem. Phys. 1986, 85, 6964. 41. Röntgen, W.K. Ann. Phys. 1892, 45, 91. 42. Bernal, S.D.; Fowler, R.H. J. Chem. Phys. 1933, 1, 515. 43. Sceats, M.G.; Rice, S.A. In Water: A Comprehensive Treatise, Franks, F.; Ed.; Plenum Press, New York, NY, 1982, Vol. 7. 44. Henn, A.R.; Kauzmann, W. J. Phys. Chem. 1989, 93, 3770. 45. Frank, H.S. In Water: A Comprehensive Treatise, Franks, F.; Ed.; Plenum Press, New York, NY, 1982, Vol. 1. 46. Blumberg, R.L.; Stanley, H.E.; Geiger, Α.; Mausbach, P. J. Chem. Phys. 1984, 80, 5230. 47. Nemethy, G.; Scheraga, H.A. J. Chem. Phys. 1962, 36, 3382. 48. Gill, S.J.; Dec, S.F.; Olofsson, G.; Wadsö, I. J. Phys. Chem. 1985, 89, 3758. 49. Grunwald, E. J. Phys. Chem. 1986, 108, 5819. 50. Benson, S.W.; Siebert, E.D. J. Am. Chem. Soc. 1992, 114, 4270. 51. G. Corongiu and E. Clementi Chem. Phys. Letters 1993, 214, 367. 52. Estrin, D.; Corongiu, G.; Clementi, E.; In Methods and Techniques in Computational Chemistry, Clementi, E.; Ed., Stef, Cagliari, Italy (1993). RECEIVED July 25, 1994

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

Chapter 8

The Dissociation of Water Analysis of the CF1 Central Force Model of Water 1

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Anna Nyberg and A. D. J. Haymet

School of Chemistry, University of Sydney, New South Wales 2006, Australia

The relative free energies of the solvated speciesH+(aq)andΗ O+(aq)are 3

calculated for the CF1 model of water. The calculations lead to an upperbound for the pH of CF1 water. Comparison is made with other calculations for the dissociation of water.

1

Recently we have calculated the pH of the CF1 central force model of water. The CF1 model is a slight modification of the central force model of Stillinger and 2

3

Rahman, designed to improve the pressure at 25 °C and 1.00 g c m " . For the CF1 model an upper bound to the pH is found to be 8.5±0.7. The model has a 1

dielectric constant of 69 ± 11. Our first calculation used classical mechanics, and predicted the equilibrium concentration of 'loosely solvated' (defined below) species sulting from the dissociation H 0 ( ^ ^ 2

and O H ^ , re­ 3

+ 0HJ" . Standard methods were Q(?)

used to calculate the relative fraction of dissociated species. Since the extent of hydration of H

+

and O H " in the CF1 model of water is not known (nor to our

knowledge, is it known for any other model), this calculation of the pH in the CF1 model established an upper bound. Further stabilisation of the 'loosely solvated' species would lead to an increase in the total equilibrium concentration of H ^ . Within the CF1 model, a hydrogen species is defined to be 'loosely solvated' if both (i) the distance to the nearest oxygen species is greater than 1.2 Â and (ii) the distance to the nearest hydrogen species is greater than 1.8 Â. 1

Corresponding author

0097-6156/94/0568-0110S08.00/0 © 1994 American Chemical Society

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

8. NYBERG AND HAYMET

The Dissociation of Water

111

Here we begin the calculation of the relative stability of tightly solvated species such as H ^ , H 0 ^ , ... , Η 0 ί ( ) and similar O H ^ species. For the 3

ç )

9

αί7

5

CF1 model, 'tightly solvated ions have all oxygen and hydrogen species connected by 'bonds', where an oxygen-hydrogen bond has a separation less than 1.2 Â, and a hydrogen-hydrogen bond has a separation less than 1.8 Â. Our ultimate goal is to calculate the total concentration of dissociated species. We begin with the H 0t Downloaded by UNIV OF GUELPH LIBRARY on October 26, 2012 | http://pubs.acs.org Publication Date: September 29, 1994 | doi: 10.1021/bk-1994-0568.ch008

3

ion, defined in the CF1 model to have all three O H distance less than 1.2

g )

Â, and all three intramolecular hydrogen-hydrogen distances less than 1.8 Â.

C F 1 M o d e l of W a t e r Despite the well-known role of pH on the structure of proteins and activity of en­ zymes, only modest interest has been shown in the dissociation of water, with the 4

5

6

notable exceptions of work by Stillinger, Warshel, and Bratos and co-workers. " The rigid models of water

10,11

9

used most frequently in computer simulations have

+

zero H ion concentration, since by construction they cannot address dissociation. 12

Some flexible models also do not permit dissociation. ""

15

2,16 17

The central force model of Stillinger and co-workers ' consists of three pair potentials acting between fractionally charged hydrogen and oxygen species. There is a single Hamiltonian which describes both intra- and inter-molecular degrees of freedom. The central force (CF) potential energies, as revised in 1978 2

by Stillinger and Rahman are:

V

o o ( r )

VWr) H

H

l

l

=

=

3

^ 0

3

} r

, V (r)

τ /

λ

ou

=

4

5

+

?6^^_ ^-4(r^,»_ 0

—

4^

i Α

1 + 40(r-2.05C ) e

72.269 6.23403 — + 3ΓΠ^ 1912

-...(r-«,»

-7.62177(r-1.45251)*

7 ί

0 - 2 f c

β

2

10 γ _j_ 4 0 ( r - 1 . 0 5 ) e

4 ^ - f 5.49305(r-2.2) > e

where the values of the constants are C\ = C2 = 1. The hydrogen species have a fraction charge of approximately one-third, which should be regarded as an effective value arising from integrating out the many contributions to the total potential energy omitted from a two-body prescription. At the temperature Τ = 3

25 °C and density 1.00 g c m " , this model has a pressure of 3,540 bar, thousands

In Structure and Reactivity in Aqueous Solution; Cramer, C., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1994.

112

STRUCTURE AND REACTIVITY IN AQUEOUS SOLUTION

18

1

of times the correct value. The CF1 model attempts to both preserve the useful properties of this model and correct the pressure. The same potential energies are used, but with the slightly different constants C\ = 0.9 and C2 = 1/1.025. With this change, the pressure for the CF1 model is 120 bar. r

Two high peaks in CF1 pair correlation functions gou( ) and gnn(r) cor­ respond to species within the same molecule, and at 25 °C, these peaks arise Downloaded by UNIV OF GUELPH LIBRARY on October 26, 2012 | http://pubs.acs.org Publication Date: September 29, 1994 | doi: 10.1021/bk-1994-0568.ch008

solely from intra-molecular correlations. The CF1 model has also been studied by integral equation methods.

19,20

The Free Energy Calculation +

+

The Helmholtz free energy of solvation for the species H , OH~ and H 0 , all of 3

importance in the dissociation of water, are calculated using gradual changes of the interaction potential between the species and the surrounding solvent. This method is called thermodynamic integration, and it has been used in calculations of the chemical potential of water

21,22

and free energy of hydration of molecules 22

and ions by Jorgensen, Kollman and others. "

25

A parameter λ describes the path chosen between the initial and final state, and the change in free energy is N

N

dH(p ,q ,\)

(2)

where Η is the Hamiltonian of a system of Ν particles, and q are coordinates and p the corresponding momenta. The angle brackets denote an average over phase space, which is approximated by a (relatively short) time average from a molecular dynamics simulation. The Helmholtz free energy corresponds to the canonical ensemble, a choice that is implemented easily in molecular dynamics simulations. The path is chosen so that no phase transitions are encountered. For example, the pressure remains positive throughout the simulation. The computational details are the same as those used earlier.

1

The change in free energy is calculated for the following processes: +

Process 1. Η ( ) —> M α